Triangle element division method, modeling apparatus, computer readable medium storing program, and data defining two-dimensional planar figure

ABSTRACT

A triangle element division method includes acquiring position information of a plurality of feature points positioned on a border of a two-dimensional planar figure, segmenting a virtual plane including the two-dimensional planar figure into a plurality of quadrangular cells, disposing additional points at positions where edges as line segments connecting two adjacent feature points on the border of the two-dimensional planar figure and borders of the plurality of cells cross, and vertices of the plurality of cells, for each of the plurality of cells, generating a plurality of triangle elements with the feature points and the additional points in the cell as vertices such that conditions that a region in the cell is filled with a plurality of the triangle elements and the triangle elements do not overlap each other are satisfied, and removing the triangle elements positioned outside the two-dimensional planar figure.

RELATED APPLICATIONS

The content of Japanese Patent Application No. 2019-194482, on the basisof which priority benefits are claimed in an accompanying applicationdata sheet, is in its entirety incorporated herein by reference.

BACKGROUND Technical Field

Certain embodiments of the present invention relate to a triangleelement division method, a modeling apparatus, a computer readablemedium storing a program, and data defining two-dimensional planarfigure.

Description of Related Art

A method of dividing a two-dimensional planar figure, of which the shapeis defined by two-dimensional point group data, into a plurality oftriangle elements has been studied since long ago as a method ofdividing a polygon into a plurality of triangles. In recent years, withdevelopment of three-dimensional measuring instrument, there is anincreasing demand for triangulating (polygonizing) a shape of ameasurement target object defined by point group data obtained bymeasurement and storing the shape of the measurement target object in acomputer.

A method of dividing a two-dimensional planar figure into a plurality oftriangle elements has been described in the related art. In a method ofthe related art, all points are connected by line segments, andoverlapping line segments are removed. In an ear slicing method of therelated art, a triangle element (ear) where two sides are sides of apolygon and remaining one side is positioned inside the polygon issliced. Another method of the related art is a method of uniquelydeciding a convex polyhedron including a point group, and positivelycreates an element close to an equilateral triangle.

SUMMARY

According to an embodiment of the invention, there is provided atriangle element division method including:

acquiring position information of a plurality of feature pointspositioned on a border of a two-dimensional planar figure;

segmenting a virtual plane including the two-dimensional planar figureinto a plurality of quadrangular cells;

disposing additional points at positions where edges as line segmentsconnecting two adjacent feature points on the border of thetwo-dimensional planar figure and borders of the plurality of cellscross, and vertices of the plurality of cells;

for each of the plurality of cells, generating a plurality of triangleelements with the feature points and the additional points in the cellas vertices such that conditions that a region in the cell is filledwith a plurality of the triangle elements and the triangle elements donot overlap each other are satisfied; and

removing the triangle elements positioned outside the two-dimensionalplanar figure.

According to another embodiment of the invention, there is provided amodeling apparatus including:

an input unit that acquires position information of a plurality offeature points positioned on a border of a two-dimensional planarfigure;

a processing unit that divides the two-dimensional planar figure, whichis defined by the position information input to the input unit, into aplurality of triangle elements; and

an output unit that outputs a processing result of the processing unit,

in which the processing unit is configured to execute

processing of segmenting a virtual plane including the two-dimensionalplanar figure into a plurality of quadrangular cells,

processing of disposing additional points at positions where edges asline segments connecting two adjacent feature points on the border ofthe two-dimensional planar figure and borders of the plurality of cellscross, and vertices of the plurality of cells,

processing of, for each of the plurality of cells, generating aplurality of triangle elements with the feature points and theadditional points in the cell as vertices such that conditions that aregion in the cell is filled with a plurality of the triangle elementsand the triangle elements do not overlap each other are satisfied,

processing of removing the triangle elements positioned outside thetwo-dimensional planar figure, and

processing of outputting, to the output unit, information defining aplurality of the triangle elements remaining after the processing ofremoving the triangle elements positioned outside the two-dimensionalplanar figure.

According to still another aspect of the invention, there is provideddata defining a two-dimensional planar figure divided into a pluralityof triangle elements,

in which an inside of the two-dimensional planar figure is segmentedinto a plurality of cells by a plurality of straight lines parallel in afirst direction and a plurality of straight lines parallel in a seconddirection crossing the first direction, and each cell included in thetwo-dimensional planar figure among the plurality of cells is dividedinto two triangle elements by one diagonal line.

According to still another aspect of the invention, there is provided acomputer readable medium storing a program that causes a computer toexecute:

a function of acquiring position information of a plurality of featurepoints positioned on a border of a two-dimensional planar figure;

a function of segmenting a virtual plane including the two-dimensionalplanar figure into a plurality of quadrangular cells;

a function of disposing additional points at positions where edges asline segments connecting two adjacent feature points on the border ofthe two-dimensional planar figure and borders of the plurality of cellscross, and vertices of the plurality of cells;

a function of, for each of the plurality of cells, generating aplurality of triangle elements with the feature points and theadditional points in the cell as vertices such that conditions that aregion in the cell is filled with a plurality of the triangle elementsand the triangle elements do not overlap each other are satisfied; and

a function of removing the triangle elements positioned outside thetwo-dimensional planar figure.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a modeling apparatus according to anembodiment.

FIG. 2 is a flowchart of a triangle element division method according tothe embodiment.

FIG. 3 is a diagram showing an example of a two-dimensional planarfigure.

FIG. 4 is an enlarged view of a part of feature points disposed on anouter border and an inner border of the two-dimensional planar figure.

FIG. 5 is a diagram of a virtual plane segmented into a plurality ofcells.

FIG. 6 is a diagram showing a part of the virtual planes in a state inwhich additional points are disposed.

FIG. 7 is a diagram of the virtual plane in which triangle elements aregenerated.

FIG. 8 is a diagram showing the virtual plane in a state before thetriangle elements are removed in Step S5 (FIG. 2).

FIG. 9 is a diagram showing the virtual plane in a state in whichtriangle elements that include edges and are positioned outside thetwo-dimensional planar figure are removed.

FIG. 10 is a diagram showing the virtual plane in a state in whichtriangle elements are removed.

FIG. 11 is a diagram showing the two-dimensional planar figure in astate of being divided into a plurality of triangle elements.

FIG. 12 is a perspective view of a portion of a three-dimensional solidshape.

FIG. 13 is a diagram showing a plurality of triangle elements generatedon a part of an outer border of an upper surface and an upper surface ofthe three-dimensional solid shape without excess and deficiency.

FIG. 14A is a diagram showing two triangle elements with one additionalpoint 42 on an edge as a vertex and one triangle element with the edgeas a side, and FIG. 14B is a diagram showing triangle elements in astate in which the additional point of the edge is moved to a positionof any one of two feature points on both sides of the additional point.

FIG. 15A is a diagram showing a plurality of triangle elements with anadditional point P on an edge as a vertex, and FIG. 15B is a diagramshowing a plurality of triangle elements in a state in which theadditional point P is moved to a position of a feature point A.

FIG. 16A is a diagram showing a plurality of triangle elements with theadditional point P on the edge as a vertex, and FIG. 16B is a diagramshowing a plurality of triangle elements in a state in which theadditional point P is moved to a position of a feature point B.

DETAILED DESCRIPTION

The method of the related art is a very simple method, but in a casewhere the number of points included in the point group increases, acomputation time becomes enormous. The ear slicing method of the relatedart cannot be used in a case where a hole exists inside the polygon.That is, a target figure to be handled is limited, and robustness islow. Another method of the related art requires a lot of computationtime.

It is desirable to provide a triangle element division method with highrobustness without needing an enormous computation time.

Since each cell is divided into the triangle elements, the number ofpoint groups for generating the triangle elements decreases. As aresult, it is possible to suppress extension of a computation time. Inaddition, the triangle elements generated outside the two-dimensionalplanar figure are removed, whereby triangle element division can beperformed even on a two-dimensional planar figure having a hole inside,and a triangle element division method with high robustness is provided.

A triangle element division method and a modeling apparatus according toan embodiment will be described referring to FIGS. 1 to 11.

FIG. 1 is a block diagram of the modeling apparatus according to theembodiment. The modeling apparatus according to the embodiment includesa processing unit 20, an input unit 21, an output unit 22, and a storageunit 23.

An input of various commands to the modeling apparatus and an input ofdata defining a shape of a two-dimensional planar figure to be subjectto triangle element division, or the like are performed to the inputunit 21. The input unit 21 includes, for example, a keyboard, a touchpanel, a pointing device, a removable medium reading device, acommunication device, and the like.

The processing unit 20 executes various kinds of processing according tocommands input to the input unit 21. For example, the processing unit 20divides the two-dimensional planar figure into a plurality of triangleelements based on data defining the shape of the two-dimensional planarfigure input to the input unit 21. The processing unit 20 outputs aprocessing result to the output unit 22. The processing unit 20 isconstituted of, for example, a central processing unit (CPU) or the likeof a computer.

The storage unit 23 includes a main storage device and an auxiliarystorage device. An operating system (OS) for the operation of theprocessing unit 20, programs that are executed by the processing unit20, and the like are stored in the auxiliary storage device. Theprocessing unit 20 reads the program stored in the auxiliary storagedevice to the main storage device and executes the program stored in themain storage device. The processing unit 20 temporarily stores variouskinds of data necessary for the processing of the processing unit 20 inthe main storage device. As the auxiliary storage device, for example, ahard disk drive, a flash memory, or the like is used. As the mainstorage device, for example, a random access memory (RAM) is used.

The output unit 22 includes, for example, a display, a removable mediumwriting device, a communication device, and the like. The displaydisplays the processing result of the processing unit 20 as an image.The removable medium writing device writes the processing result of theprocessing unit 20 to a removable medium. The communication devicetransmits the processing result of the processing unit 20 to externalequipment.

FIG. 2 is a flowchart of the triangle element division method accordingto the embodiment. A function of each step of the flowchart shown inFIG. 2 is realized by the processing unit 20 (FIG. 1) executing theprogram stored in the storage unit 23. In each step of the flowchart ofFIG. 2, processing that is executed on the two-dimensional planar figureis executed by executing data processing on position data defining aposition of each point of a point group defining the shape of thetwo-dimensional planar figure.

First, in Step S1, the processing unit 20 (FIG. 1) acquires positioninformation of a plurality of feature points of the two-dimensionalplanar figure input to the input unit 21 (FIG. 1). The positioninformation includes, for example, values of an x coordinate and a ycoordinate on xy coordinates.

FIG. 3 shows an example of a two-dimensional planar FIG. 30. An outershape of the two-dimensional planar FIG. 30 is demarcated by a border31. A hole 33 is provided inside a region surrounded by the border 31. Aperiphery of the hole 33 is demarcated by a border 32. A region betweenthe outer border 31 and the inner border 32 is the inside of thetwo-dimensional planar FIG. 30. A region outside the outer border 31 anda region inside the hole 33 are the outside of the two-dimensionalplanar FIG. 30. The outer border 31 includes a portion 35 convex outwardand a portion 36 convex inward. A plurality of feature points 40 aredefined on the outer border 31 and the inner border 32. A serial numberis given to a plurality of feature points 40, and one feature point 40is specified by the serial number. Positions of a plurality of featurepoints 40 are defined by position information, and the shape of thetwo-dimensional planar figure is defined by the positions of a pluralityof feature points 40. In FIG. 3, only a part of feature points 40 isshown.

FIG. 4 is an enlarged view of a part of the feature points 40 disposedon the outer border 31 and the inner border 32. Serial numbers i−1, i,and i+1 are given to three feature points 40 disposed on the outerborder 31, and serial numbers j−1, j, and j+1 are given to three featurepoint 40 disposed on the inner border 32. Here, i and j are integers.When the feature point 40 of a greater number is viewed from the featurepoint 40 of a smaller number, left and right regions are defined as theinside and the outside of the two-dimensional planar FIG. 30,respectively. The definition of the inside and the outside may bereversed. The inside and the outside may be defined using other methods.A line segment with two feature points 40 of continuous serial numbersas both ends is referred to as an edge 41.

After Step S1, in Step S2 (FIG. 2), a finite virtual plane including thetwo-dimensional planar FIG. 30 is defined, and the virtual plane issegmented into a plurality of cells.

FIG. 5 is a diagram of a virtual plane 50 segmented into a plurality ofcells 51. The two-dimensional planar FIG. 30 is included in the virtualplane 50. The virtual plane 50 is separated by a plurality of straightlines parallel in a first direction D1 (in FIG. 5, a right-leftdirection) and a plurality of straight lines parallel in a seconddirection D2 (in FIG. 5, an up-down direction) perpendicular to thefirst direction D1 and is segmented into a plurality of cells 51. Eachof a plurality of cells 51 is, for example, a rectangle.

Next, in Step S3 (FIG. 2), additional points 42 are disposed atpositions where the edges 41 and the borders of the cells 51 cross, andvertices of the cells 51.

FIG. 6 shows a part of the virtual plane 50 in a state in which theadditional points 42 are disposed. The additional points 42 are disposedat the positions where the borders 52 of the cells 51 and the edges 41cross. The additional points 42 are also disposed at the vertices of thecells 51. In a case where a situation occurs in which the border 52 ofthe cell 51 passes through the feature point 40, the virtual plane 50and the two-dimensional planar FIG. 30 are relatively slightly shifted,thereby avoiding the occurrence of a situation in which the border 52 ofthe cell 51 passes through the feature point 40.

Next, in Step S4 (FIG. 2), a plurality of triangle elements with thefeature points 40 and the additional points 42 as vertices are generatedfor each cell 51. In this case, the triangle elements are generated suchthat two conditions that the cell 51 is completely filled with aplurality of triangle elements and the triangle elements do not overlapeach other are satisfied. Generating the triangle elements so as tosatisfy the two conditions is referred to as generating the triangleelements without excess and deficiency.

FIG. 7 shows a part of the virtual plane 50 in which the triangleelements 60 are generated. Hereinafter, a procedure for generating aplurality of triangle elements 60 will be described. First, linesegments that connect all feature points 40 in one cell 51 and alladditional points 42 positioned on the border 52 of the cell 51 aregenerated. Two line segments crossing each other among the line segmentsare detected, and one of the two line segments is deleted. The procedureis repeated until the line segments crossing each other are eliminated.With this, the triangle elements 60 are generated in the cell 51 withoutexcess and deficiency. When a plurality of line segments are generatedin order, in a case where a line segment to be newly generated willcross an already generated line segment, generation of a new linesegment may not be performed.

Next, in Step S5 (FIG. 2), the triangle elements 60 outside thetwo-dimensional planar FIG. 30 are removed. A specific procedure of StepS5 will be described referring to FIGS. 8 to 11.

FIG. 8 is a diagram showing the virtual plane 50 in a state before thetriangle elements 60 are removed in Step S5. In this stage, the triangleelements 60 are generated on both sides of each of all edges 41. Sincethe edge 41 corresponds to the border of the two-dimensional planar FIG.30, one of the triangle elements 60 on both sides of the edge 41 ispositioned inside the two-dimensional planar FIG. 30, and the othertriangle element 60 is positioned outside the two-dimensional planarFIG. 30. First, for each of all edges 41, a triangle element 60A thatincludes the edge 41 and is positioned outside the two-dimensionalplanar FIG. 30 is removed. In FIG. 8, the triangle element 60A thatincludes the edge 41 and is positioned outside the two-dimensionalplanar FIG. 30 is hatched.

FIG. 9 is a diagram showing the virtual plane 50 in a state in which thetriangle element 60A (FIG. 8) that includes the edge 41 and ispositioned outside the two-dimensional planar FIG. 30 is removed. Inthis stage, triangle elements 60B that are positioned outside thetwo-dimensional planar FIG. 30 and do not include the edge 41 remainunremoved. In FIG. 9, the triangle elements 60B are hatched.

The triangle elements 60 that are positioned inside the two-dimensionalplanar FIG. 30 abut on adjacent triangle elements 60 through sides otherthan the edge 41. That is, the triangle elements 60 are disposed on bothsides of the sides of the triangle elements 60 other than the edge 41.Accordingly, in extracting the triangle elements 60B positioned outsidethe two-dimensional planar FIG. 30, first, sides, only one side of whichhas the triangle element 60, should be extracted from among the sidesother than the edge 41. In FIG. 9, as the sides, only one side of whichhas the triangle element 60, sides 61 are extracted. Four triangleelements 60B including the sides 61 are determined to be positionedoutside the two-dimensional planar FIG. 30.

FIG. 10 is a diagram showing the virtual plane 50 in a state in whichthe triangle elements 60B (FIG. 9) are removed. The triangle elements60A (FIG. 8) and the triangle elements 60B (FIG. 9) positioned outsidethe two-dimensional planar FIG. 30 are removed, and the triangleelements 60 remain only inside the two-dimensional planar FIG. 30.

Next, in Step S6 (FIG. 2), the processing unit 20 outputs data definingthe two-dimensional planar FIG. 30, of which the inside is divided intoa plurality of triangle elements 60, to the output unit 22.

FIG. 11 is a diagram showing the two-dimensional planar FIG. 30 in astate of being divided into a plurality of triangle elements 60. Theinside of the two-dimensional planar FIG. 30 is segmented into aplurality of cells 51 by a plurality of straight lines parallel in thefirst direction D1 and a plurality of straight lines parallel in thesecond direction D2 perpendicular to the first direction D1. A pluralityof triangle elements 60 are generated for each cell 51, and the triangleelements 60 are not disposed outside the two-dimensional planar FIG. 30.Inside the two-dimensional planar FIG. 30, a plurality of triangleelements 60 are generated without excess and deficiency.

In cells 51A where at least one of the outer border 31 and the innerborder 32 of the two-dimensional planar FIG. 30 passes therethrough, aplurality of triangle elements 60C with the edges 41 as at least oneside are generated. Cells 51B of which the entire region is included inthe two-dimensional planar FIG. 30 are divided into two triangleelements 60D by one diagonal line. Even in the cells 51A where theborders 31 and 32 of the two-dimensional planar FIG. 30 passtherethrough, one triangle element 60E divided by a diagonal line of thecell 51 may be generated.

Next, excellent effects of the above-described embodiment will bedescribed.

In the above-described embodiment, since the two-dimensional planar FIG.30 is segmented into a plurality of cells 51 (Step S2), and each cell 51is divided into the triangle elements 60 (Step S4), the number offeature points 40 to be focused at the time of generating the triangleelements 60 is reduced. In addition, it is possible to executeprocessing of dividing each cell 51 into the triangle elements 60 inparallel. With this, it is possible to reduce a time necessary forcomputation.

In the above-described embodiment, since the triangle elements 60generated outside the two-dimensional planar FIG. 30 are removed (StepS5), even the two-dimensional planar FIG. 30 having the hole 33 (FIG. 3)inside can be divided into a plurality of triangle elements 60. For thisreason, it can be said that the triangle element division methodaccording to the above-described embodiment has high robustness.

An algorithm of the triangle element division method according to theabove-described embodiment is simple. For this reason, an excellenteffect that, in a case where a trouble occurs in a division result,search for a cause of the trouble is easily performed is obtained.

Next, a modification example of the above-described embodiment will bedescribed. In the above-described embodiment, the virtual plane 50 (FIG.5) is separated by two sets of straight line groups perpendicular toeach other to be segmented into a plurality of cells 51. That is, thefirst direction D1 and the second direction D2 are perpendicular. Forthis reason, the cells 51 are squares or rectangles. As another method,the first direction D1 and the second direction D2 may cross obliquely.In this case, the cells 51 are parallelograms. The virtual plane 50 maybe segmented such that the cells 51 are quadrangles other thanparallelograms.

Next, a triangle element division method according to another embodimentwill be described referring to FIGS. 12 to 16B. In the triangle elementdivision method according to the embodiment shown in FIGS. 1 to 11, thesingle two-dimensional planar FIG. 30 (FIG. 3 and the like) is dividedinto the triangle elements 60 without excess and deficiency. In thefollowing embodiment, the two-dimensional planar FIG. 30 constitutes onesurface of a three-dimensional solid shape.

FIG. 12 is a perspective view of a portion of a three-dimensional solidshape 70. Apart (hereinafter, referred to as an upper surface 71) of asurface of the three-dimensional solid shape 70 has the same shape asthe two-dimensional planar FIG. 30 (FIG. 3). That is, the upper surface71 has an outer border 31 and an inner border 32 similarly to thetwo-dimensional planar FIG. 30. An outer side surface 72 and an innerside surface 73 of the three-dimensional solid shape 70 continue withthe outer border 31 and the inner border 32 of the upper surface 71,respectively. In the embodiment, it is possible to achieve consistencybetween a plurality of triangle elements 60 generated on the uppersurface 71 and a plurality of triangle elements generated on the outerside surface 72 and the inner side surface 72. Here, “achieveconsistency” means that the triangle elements on one surface between twosurfaces connected through the borders 31 and 32 and the triangleelements on the other surface are made to have a relationshipcorresponding to each other on a one-to-one basis through the edges.

First, the triangle element division method according to the embodimentshown in FIGS. 1 to 11 is applied to divide the upper surface 71 into aplurality of triangle elements 60 without excess and deficiency. In FIG.12, only a part of triangle elements 60 is shown.

FIG. 13 shows a plurality of triangle elements 60 generated on a part ofthe outer border 31 of the upper surface 71 and the upper surface 71without excess and deficiency. In addition, triangle elements 65generated on the outer side surface 72 without excess and deficiency areshown. In dividing the upper surface 71 into a plurality of triangleelements 60, in a case where the method shown in FIGS. 1 to 11 isapplied, an additional point 42 are disposed on the outer border 31 inaddition to feature points 40. In the upper surface 71, two triangleelements 60F with the additional point 42 as a vertex are generated.

In a case where processing of dividing the outer side surface 72 into aplurality of triangle elements 65 is executed in parallel withprocessing of dividing the upper surface 71 into a plurality of triangleelements 60, in dividing the outer side surface 72 into a plurality oftriangle elements 65, the additional point 42 is not disposed on theouter border 31. In a case where the outer side surface 72 is dividedinto a plurality of triangle elements 65 in this state, two triangleelements 60F with the additional point 42 as a vertex are generated onthe upper surface 71 with respect to one triangle element 65F that isgenerated on the outer side surface 72. In this state, it cannot be saidthat consistency is achieved.

Next, a method of avoiding the generation of triangle elements having aninconsistent positional relationship, such as the two triangle elements60F and the one triangle element 65F shown in FIG. 13, will be describedreferring to FIGS. 14A and 14B.

FIG. 14A is a diagram showing two triangle elements 60F with oneadditional point 42 on the edge 41 as a vertex and one triangle element65F with the edge 41 as a side. Feature points 40 are disposed on bothsides of the additional point 42 on the edge 41. Each of the triangleelements 60F has one additional point 42 on the edge 41 and one featurepoint 40 on the edge 41 as two vertices. The additional point 42 on theedge 41 is moved to a position of any one of the two feature points 40on both sides thereof.

FIG. 14B is a diagram showing triangle elements in a state in which theadditional point 42 on the edge 41 is moved to the position of any oneof the two feature points 40 on both sides thereof. With the movement,the additional point 42 disappears, and the shape of the triangleelement 60F with the additional point 42 as a vertex changes.Specifically, one of the two triangle elements 60F disappears, and theother triangle element 60F becomes large. In this way, the additionalpoint 42 on the edge 41 is moved to the position of one of the featurepoints 40 positioned at both ends of the edge 41, whereby it is possibleto remove the additional point 42 on the edge 41.

In the example shown in FIGS. 14A and 14B, even though the additionalpoint 42 is moved to the position of any feature point 40 on both sidesthereof, the same effect is obtained. Incidentally, a result to beobtained may not satisfy a condition “the triangle elements aregenerated without excess and deficiency” depending on a direction ofmoving the additional point 42. Hereinafter, an example where thecondition “the triangle elements are generated without excess anddeficiency” is not satisfied will be described referring to FIGS. 15Aand 16B.

FIG. 15A is a diagram showing a plurality of triangle elements 60 withan additional point 42 on the edge 41 as a vertex. The additional point42 is denoted as P, feature points 40 on both sides of the additionalpoint 42 are denoted as A and B, respectively, and five other featurepoints 40 that are connected to the additional point 42 by the sides ofthe triangle elements 60 are denoted as C, D, E, F, and G, respectively.Six triangle elements GAP, GPF, FPE, EPD, DPC, and CPB in total aregenerated.

FIG. 15B is a diagram showing a plurality of triangle elements 60 in astate in which the additional point P is moved to a position of thefeature point A. The original triangle element GAP disappears. In thisstate, the triangle element CPB overlaps other triangle elements GPF,FPE, EPD, and DPC. Accordingly, the condition “generation without excessand deficiency” is not satisfied. In this case, a sum of areas of thefive triangle elements GPF, FPE, EPD, DPC, and CPB remaining after theadditional point P is moved is greater than a sum of areas of theoriginal six triangle elements GAP, GPF, FPE, EPD, DPC, and CPB.

FIG. 16A is a diagram showing a plurality of triangle elements 60 in thesame state as the state of FIG. 15A before the additional point P ismoved. FIG. 16B is a diagram showing a plurality of triangle elements 60in a state in which the additional point P is moved to the feature pointB. In this case, the original triangle element CPB disappears, and theremaining five triangle elements GAP, GPF, FPE, EPD, and DPC remain. Theremaining five triangle elements GAP, GPF, FPE, EPD, and DPC satisfy thecondition “generation without excess and deficiency”. In this case, asum of areas of the five triangle elements GAP, GPF, FPE, EPD, and DPCremaining after the additional point P is moved is equal to the sum ofthe areas of the original six triangle elements GAP, GPF, FPE, EPD, DPC,and CPB.

After the additional point 42 on the edge 41 is moved to the position ofany one of the feature points 40 at both ends of the edge 41, in a casewhere the sum of the areas of a plurality of triangle elements 60 withthe additional point 42 as a vertex is greater than the sum of the areasof the triangle elements before the movement (FIG. 15B), determinationcan be made that the movement is inappropriate movement. In this case,as shown in FIG. 16B, the additional point 42 should be moved to theposition of the opposite feature point 40 on the edge 41.

Next, excellent effects of the embodiment shown in FIGS. 12 to 16B willbe described.

In the embodiment, after a portion, which is a plane, of the surface ofthe three-dimensional solid shape 70 (FIG. 12) is divided into aplurality of triangle elements 60, as shown in FIGS. 14B and 16B, theadditional point 42 (FIG. 14A) is not disposed on the border 31 or 32 ofthe region divided into a plurality of triangle elements 60. For thisreason, in a case where another surface region connected to a surfaceregion divided into a plurality of triangle elements 60 through theborder is divided into a plurality of triangle elements, it is possibleto achieve consistency of arrangement of the triangle elements of thetwo surface regions connected through the border.

It should be understood that the above-described embodiments areexamples and partial replacement or a combination of the configurationsshown indifferent embodiments is possible. The same advantageous effectby the same configuration in a plurality of embodiments is notconsecutively mentioned for each embodiment. In addition, the inventionis not limited to the above-described embodiments. For example, it willbe obvious to those skilled in the art that various alterations,improvements, combinations, and the like can be made.

It should be understood that the invention is not limited to theabove-described embodiment, but may be modified into various forms onthe basis of the spirit of the invention. Additionally, themodifications are included in the scope of the invention.

What is claimed is:
 1. A triangle element division method comprising:acquiring position information of a plurality of feature pointspositioned on a border of a two-dimensional planar figure; segmenting avirtual plane including the two-dimensional planar figure into aplurality of quadrangular cells; disposing additional points atpositions where edges as line segments connecting two adjacent featurepoints on the border of the two-dimensional planar figure and borders ofthe plurality of cells cross, and vertices of the plurality of cells;for each of the plurality of cells, generating a plurality of triangleelements with the feature points and the additional points in the cellas vertices such that conditions that a region in the cell is filledwith a plurality of the triangle elements and the triangle elements donot overlap each other are satisfied; and removing the triangle elementspositioned outside the two-dimensional planar figure.
 2. The triangleelement division method according to claim 1, wherein the removing ofthe triangle elements positioned outside the two-dimensional planarfigure has removing the triangle elements outside the two-dimensionalplanar figure among the triangle elements positioned on both sides ofthe edges, and repeating a procedure for removing the triangle elementswith a side, only one side of which has the triangle element, amongsides other than the edges until the triangle elements with the side,only one side of which has the triangle element, are eliminated.
 3. Thetriangle element division method according to claim 1, furthercomprising: after the removing of the triangle elements positionedoutside the two-dimensional planar figure, moving the additional pointon the edge to a position of the feature point in an end portion of theedge and deforming the triangle elements with the moved additional pointas a vertex based on the movement of the additional point.
 4. Thetriangle element division method according to claim 3, wherein, inmoving the additional point on the edge, the additional point is movedin a direction in which a sum of areas of a plurality of the triangleelements with the moved additional point as a vertex does not increasebefore and after the movement.
 5. A modeling apparatus comprising: aninput unit that acquires position information of a plurality of featurepoints positioned on a border of a two-dimensional planar figure; aprocessing unit that divides the two-dimensional planar figure, which isdefined by the position information input to the input unit, into aplurality of triangle elements; and an output unit that outputs aprocessing result of the processing unit, wherein the processing unit isconfigured to execute processing of segmenting a virtual plane includingthe two-dimensional planar figure into a plurality of quadrangularcells, processing of disposing additional points at positions whereedges as line segments connecting two adjacent feature points on theborder of the two-dimensional planar figure and borders of the pluralityof cells cross, and vertices of the plurality of cells, processing of,for each of the plurality of cells, generating a plurality of triangleelements with the feature points and the additional points in the cellas vertices such that conditions that a region in the cell is filledwith a plurality of the triangle elements and the triangle elements donot overlap each other are satisfied, processing of removing thetriangle elements positioned outside the two-dimensional planar figure,and processing of outputting, to the output unit, information defining aplurality of the triangle elements remaining after the processing ofremoving the triangle elements positioned outside the two-dimensionalplanar figure.
 6. The modeling apparatus according to claim 5, whereinthe processing unit is configured to, in the processing of removing thetriangle elements positioned outside the two-dimensional planar figure,execute processing of removing the triangle elements outside thetwo-dimensional planar figure among the triangle elements positioned onboth sides of the edges, and processing of repeating a procedure forremoving the triangle elements with a side, only one side of which hasthe triangle element, among sides other than the edges of the triangleelements until the triangle elements with the side, only one side ofwhich has the triangle element, are eliminated.
 7. The modelingapparatus according to claim 5, wherein the processing unit isconfigured to, after the processing of removing the triangle elementspositioned outside the two-dimensional planar figure, further executeprocessing of moving the additional point on the edge to a position ofthe feature point in an end portion of the edge and deforming thetriangle elements with the moved additional point as a vertex based onthe movement of the additional point.
 8. The modeling apparatusaccording to claim 7, wherein the processing unit is configured to, inthe processing of moving the additional point on the edge, move theadditional point in a direction in which a sum of areas of a pluralityof the triangle elements with the moved additional point as a vertexdoes not increase before and after the movement.
 9. Data defining atwo-dimensional planar figure divided into a plurality of triangleelements, wherein an inside of the two-dimensional planar figure issegmented into a plurality of cells by a plurality of straight linesparallel in a first direction and a plurality of straight lines parallelin a second direction crossing the first direction, and each cellincluded in the two-dimensional planar figure among the plurality ofcells is divided into two triangle elements by one diagonal line.
 10. Acomputer readable medium storing a program that causes a computer toexecute: a function of acquiring position information of a plurality offeature points positioned on a border of a two-dimensional planarfigure; a function of segmenting a virtual plane including thetwo-dimensional planar figure into a plurality of quadrangular cells; afunction of disposing additional points at positions where edges as linesegments connecting two adjacent feature points on the border of thetwo-dimensional planar figure and borders of the plurality of cellscross, and vertices of the plurality of cells; a function of, for eachof the plurality of cells, generating a plurality of triangle elementswith the feature points and the additional points in the cell asvertices such that conditions that a region in the cell is filled with aplurality of the triangle elements and the triangle elements do notoverlap each other are satisfied; and a function of removing thetriangle elements positioned outside the two-dimensional planar figure.